#### Chapters

Chapter 2: Complex Numbers

Chapter 3: Theory of Equations

Chapter 4: Inverse Trigonometric Functions

Chapter 5: Two Dimensional Analytical Geometry-II

Chapter 6: Applications of Vector Algebra

Chapter 7: Applications of Differential Calculus

Chapter 8: Differentials and Partial Derivatives

Chapter 9: Applications of Integration

Chapter 10: Ordinary Differential Equations

Chapter 11: Probability Distributions

Chapter 12: Discrete Mathematics

## Solutions for Chapter 3: Theory of Equations

Below listed, you can find solutions for Chapter 3 of Tamil Nadu Board of Secondary Education Tamil Nadu Board Samacheer Kalvi for Class 12th Mathematics Volume 1 and 2 Answers Guide.

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 3 Theory of Equations Exercise 3.1 [Pages 106 - 107]

If the sides of a cubic box are increased by 1, 2, 3 units respectively to form a cuboid, then the volume is increased by 52 cubic units. Find the volume of the cuboid

Construct a cubic equation with roots 1, 2 and 3

Construct a cubic equation with roots 1, 1, and – 2

Construct a cubic equation with roots `2, 1/2, and 1`

If α, β and γ are the roots of the cubic equation x^{3} + 2x^{2} + 3x + 4 = 0, form a cubic equation whose roots are 2α, 2β, 2γ

If α, β and γ are the roots of the cubic equation x^{3} + 2x^{2} + 3x + 4 = 0, form a cubic equation whose roots are `1/alpha, 1/beta, 1/γ`

If α, β and γ are the roots of the cubic equation x^{3} + 2x^{2} + 3x + 4 = 0, form a cubic equation whose roots are `- alpha, -beta, -γ`

Solve the equation 3x^{3} – 16x^{2 }+ 23x – 6 = 0 if the product of two roots is 1

Find the sum of squares of roots of the equation `2x^4 - 8x^3 + 6x^2 - 3` = 0

Solve the equation x^{3} – 9x^{2} + 14x + 24 = 0 if it is given that two of its roots are in the ratio 3 : 2

If α, β, and γ are the roots of the polynomial equation ax^{3} + bx^{2} + cx + d = 0, find the value of `sum alpha/(betaγ)` in terms of the coefficients

If α, β, γ and δ are the roots of the polynomial equation 2x^{4} + 5x^{3} – 7x^{2} + 8 = 0, find a quadratic equation with integer coefficients whose roots are α + β + γ + δ and αβγδ

If p and q are the roots of the equation lx^{2} + nx + n = 0, show that `sqrt("p"/"q") + sqrt("q"/"p") + sqrt("n"/l)` = 0

If the equations x^{2} + px + q = 0 and x^{2} + p’x + q’ = 0 have a common root, show that it must be equal to `("pq'" - "p'q")/("q" - "q")` or `("q" - "q'")/("p'" - "P")`

A 12 metre tall tree was broken into two parts. It was found that the height of the part which was left standing was the cube root of the length of the part that was cut away. Formulate this into a mathematical problem to find the height of the part which was left standing

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 3 Theory of Equations Exercise 3.2 [Page 112]

If k is real, discuss the nature of the roots of the polynomial equation 2x^{2} + kx + k = 0, in terms of k

Find a polynomial equation of minimum degree with rational coefficients, having `2 + sqrt(3)"i"` as a root

Find a polynomial equation of minimum degree with rational coefficients, having 2i + 3 as a root

Find a polynomial equation of minimum degree with rational coefficients, having `sqrt(5) - sqrt(3)` as a root

Prove that a straight line and parabola cannot intersect at more than two points

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 3 Theory of Equations Exercise 3.3 [Page 117]

Solve the cubic equation: 2x^{3} – x^{2} – 18x + 9 = 0 if sum of two of its roots vanishes

Solve the equation 9x^{3} – 36x^{2} + 44x – 16 = 0 if the roots form an arithmetic progression

Solve the equation 3x^{3} – 26x^{2} + 52x – 24 = 0 if its roots form a geometric progression

Determine k and solve the equation 2x^{3} – 6x^{2} + 3x + k = 0 if one of its roots is twice the sum of the other two roots

Find all zeros of the polynomial x^{6} – 3x^{5} – 5x^{4} + 22x^{3} – 39x^{2} – 39x + 135, if it is known that 1 + 2i and `sqrt(3)` are two of its zeros

Solve the cubic equations:

2x^{3} – 9x^{2} + 10x = 3

Solve the cubic equations:

8x^{3} – 2x^{2} – 7x + 3 = 0

Solve the equation:

x^{4} – 14x^{2} + 45 = 0

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 3 Theory of Equations Exercise 3.4 [Page 118]

Solve: (x – 5)(x – 7) (x + 6)(x + 4) = 504

Solve: (x – 4)(x – 2)(x- 7)(x + 1) = 16

Solve: (2x – 1)(x + 3)(x – 2)(2x + 3) + 20 = 0

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 3 Theory of Equations Exercise 3.5 [Page 124]

Solve the following equations

sin² x – 5 sin x + 4 = 0

Solve the following equations

12x^{3} + 8x = 29x^{2} – 4 = 0

Examine for the rational roots of

2x^{3} – x^{2} – 1 = 0

Examine for the rational roots of

x^{8} – 3x + 1 = 0

Solve: `8x^(3/(2"n")) - 8x^((-3)/(2"n"))` = 63

Solve: `2sqrt(x/"a") + 3sqrt("a"/x) = "b"/"a" + (6"a")/"b"`

Solve the equation

6x^{4} – 35x^{3} + 62x^{2} – 35x + 6 = 0

Solve the equation

x^{4} + 3x^{3} – 3x – 1 = 0

Find all real numbers satisfying 4^{x} – 3(2^{x+2}) + 2^{5} = 0

Solve the equation 6x^{4} – 5x^{3} – 38x^{2} – 5x + 6 = 0 if it is known that `1/3` is a solution

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 3 Theory of Equations Exercise 3.6 [Page 127]

Discuss the maximum possible number of positive and negative roots of the polynomial equation 9x^{9} – 4x^{8} + 4x^{7} – 3x^{6} + 2x^{5} + x^{3} + 7x^{2} + 7x + 2 = 0

Discuss the maximum possible number of positive and negative roots of the polynomial equations x^{2} – 5x + 6 and x^{2} – 5x + 16. Also, draw a rough sketch of the graphs

Show that the equation x^{9} – 5x^{5} + 4x^{4} + 2x^{2} + 1 = 0 has atleast 6 imaginary solutions

Determine the number of positive and negative roots of the equation x^{9} – 5x^{8} – 14x^{7} = 0

Find the exact number of real zeros and imaginary of the polynomial x^{9} + 9x^{7} + 7x^{5} + 5x^{3} + 3x

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 3 Theory of Equations Exercise 3.7 [Pages 127 - 128]

#### MCQ

Choose the correct alternative:

A zero of x^{3} + 64 is

0

4

4i

– 4

Choose the correct alternative:

If f and g are polynomials of degrees m and n respectively, and if h(x) = (f o g)(x), then the degree of h is

mn

m + n

m

^{n}n

^{m}

Choose the correct alternative:

A polynomial equation in x of degree n always has

n distinct roots

n real roots

n complex roots

at most one root

Choose the correct alternative:

If α, β and γ are the zeros of x^{3} + px^{2} + qx + r, then `sum 1/alpha` is

`- "q"/"r"`

`- "p"/"r"`

`"q"/"r"`

`-"q"/"p"`

Choose the correct alternative:

According to the rational root theorem, which number is not possible rational root of 4x^{7} + 2x^{7} – 10x^{3} – 5?

– 1

`5/4`

`4/5`

5

Choose the correct alternative:

The polynomial x^{3} – kx^{2} + 9x has three real roots if and only if, k satisfies

|k| ≤ 6

k = 0

|k| > 6

|k| ≥ 6

Choose the correct alternative:

The number of real numbers in [0, 2π] satisfying sin^{4}x – 2 sin^{2}x + 1 is

2

4

1

`oo`

Choose the correct alternative:

If x^{3} + 12x^{2} + 10ax + 1999 definitely has a positive zero, if and only if

a ≥ 0

a > 0

a < 0

a ≤ 0

Choose the correct alternative:

The polynomial x^{3} + 2x + 3 has

one negative and two imaginary zeros

one positive and two imaginary zeros

three real zeros

no zeros

Choose the correct alternative:

The number of positive roots of the polynomials `sum_("j" = 0)^"n" ""^"n""C"_"r" (- 1)^"r" x^"r"` is

0

n

< n

r

## Solutions for Chapter 3: Theory of Equations

## Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 3 - Theory of Equations

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Concepts covered in Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 3 Theory of Equations are Introduction to Theory of Equations, Basics of Polynomial Equations, Vieta’s Formulae and Formation of Polynomial Equations, Nature of Roots and Nature of Coefficients of Polynomial Equations, Roots of Higher Degree Polynomial Equations, Polynomial Equations with No Additional Information, Polynomials with Additional Information, Descartes Rule.

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